Black-box Optimization Research Articles

A novel inverse methodology for the extraction of bulk elasto-plastic tensile properties of metals using spherical instrumented indentation

This paper presents an inverse methodology relying on finite element simulation based optimization to extract material constitutive parameters from a single spherical indentation, which are used to estimate its bulk true stress–true strain curve. Methods of this type usually assume that the tensile curve of the studied material can be modelled by a power-law involving, for example, Young’s modulus, yield stress and a hardening coefficient. Since the tensile behaviour of some materials cannot be adequately fitted by such models, our approach rather relies on an optimization procedure which extracts a group of six points on the true stress–true strain curve, distributed as per a geometric progression in the strain space, as well as the elastic modulus. A direct search black-box optimization algorithm is used and is shown to be capable of eluding local minima. A surrogate step is introduced in the methodology, which is a simplified version of the inverse problem, to find a suitable starting point. The performance of the method is first investigated in a numerical study in which the estimated true stress–true strain curves lie within a maximum error of 5.7% from the corresponding target tensile curves, for four materials with different hardening behaviours. The maximum errors on the extracted elastic modulus and yield stress are 0.5% and 11.1%, respectively. A study of the convergence behaviour of the proposed method demonstrates an expected significant increase in computational times as compared to when using a hardening model. Finally, an experimental application of the method to two materials with a plastic plateau leads to true stress–true strain curves estimations with average differences of 3.9% and 0.9% with the macroscopic experimental tensile curve, respectively. The method is thus found to be accurate even in the presence of modelling and experimental errors.

Optimal sizing of photovoltaic-wind-diesel-battery power supply for mobile telephony base stations

The paper proposes a novel planning approach for optimal sizing of standalone photovoltaic-wind-diesel-battery power supply for mobile telephony base stations. The approach is based on integration of a comprehensive probabilistic sequential Monte Carlo simulator and a black-box optimizer using DIRECT (DIviding RECTangles) method. The main property of the simulator is that input variables are modeled as correlated random processes rather than independent random variables without a time index. By taking into account autocorrelation and mutual correlation, temporal properties of all input variables are kept realistic, such as sunrise and sunset times in irradiance model, daily and seasonal changes in temperature model, and consumption of cooling devices and electronic equipment which depends on the ambient temperature and setting of the parameters of the control devices. The optimization target is to select rated capacities of major system components and to tune the main control parameters for achieving minimum total annual costs without compromising system reliability. The proposed algorithm is tested on planning a typical 2 kW potential base station located on a windy and sunny hill in the Mediterranean region.

Output Space Entropy Search Framework for Multi-Objective Bayesian Optimization

We consider the problem of black-box multi-objective optimization (MOO) using expensive function evaluations (also referred to as experiments), where the goal is to approximate the true Pareto set of solutions by minimizing the total resource cost of experiments. For example, in hardware design optimization, we need to find the designs that trade-off performance, energy, and area overhead using expensive computational simulations. The key challenge is to select the sequence of experiments to uncover high-quality solutions using minimal resources. In this paper, we propose a general framework for solving MOO problems based on the principle of output space entropy (OSE) search: select the experiment that maximizes the information gained per unit resource cost about the true Pareto front. We appropriately instantiate the principle of OSE search to derive efficient algorithms for the following four MOO problem settings: 1) The most basic single-fidelity setting, where experiments are expensive and accurate; 2) Handling black-box constraints which cannot be evaluated without performing experiments; 3) The discrete multi-fidelity setting, where experiments can vary in the amount of resources consumed and their evaluation accuracy; and 4) The continuous-fidelity setting, where continuous function approximations result in a huge space of experiments. Experiments on diverse synthetic and real-world benchmarks show that our OSE search based algorithms improve over state-of-the-art methods in terms of both computational-efficiency and accuracy of MOO solutions.

Uncertainty quantification and exploration–exploitation trade-off in humans

The main objective of this paper is to outline a theoretical framework to analyse how humans’ decision-making strategies under uncertainty manage the trade-off between information gathering (exploration) and reward seeking (exploitation). A key observation, motivating this line of research, is the awareness that human learners are amazingly fast and effective at adapting to unfamiliar environments and incorporating upcoming knowledge: this is an intriguing behaviour for cognitive sciences as well as an important challenge for Machine Learning. The target problem considered is active learning in a black-box optimization task and more specifically how the exploration/exploitation dilemma can be modelled within Gaussian Process based Bayesian Optimization framework, which is in turn based on uncertainty quantification. The main contribution is to analyse humans’ decisions with respect to Pareto rationality where the two objectives are improvement expected and uncertainty quantification. According to this Pareto rationality model, if a decision set contains a Pareto efficient (dominant) strategy, a rational decision maker should always select the dominant strategy over its dominated alternatives. The distance from the Pareto frontier determines whether a choice is (Pareto) rational (i.e., lays on the frontier) or is associated to “exasperate” exploration. However, since the uncertainty is one of the two objectives defining the Pareto frontier, we have investigated three different uncertainty quantification measures and selected the one resulting more compliant with the Pareto rationality model proposed. The key result is an analytical framework to characterize how deviations from “rationality” depend on uncertainty quantifications and the evolution of the reward seeking process.

Joint Power and Channel Optimization of Agricultural Wireless Sensor Networks Based on Hybrid Deep Reinforcement Learning

The reduction of maintenance costs in agricultural wireless sensor networks (WSNs) requires reducing energy consumption. At the same time, care should be taken not to affect communication quality and network lifetime. This paper studies a joint optimization algorithm for transmitted power and channel allocation based on deep reinforcement learning. First, an optimization model to measure network reward was established under the constraint of the signal-to-interference plus-noise-ratio (SINR) threshold, which includes continuous power variables and discrete channel variables. Secondly, considering the dynamic changes of agricultural WSNs, the network control is described as a Markov decision process with continuous state and action space. A deep deterministic policy gradient (DDPG) reinforcement learning scheme suitable for mixed variables was designed. This method could obtain a control scheme that maximizes network reward by means of black-box optimization for continuous transmitted power and discrete channel allocation. Experimental results indicated that the studied algorithm has stable convergence. Compared with traditional protocols, it can better control the transmitted power and allocate channels. The joint power and channel optimization provides a reference solution for constructing an energy-balanced network.

Online learning for hierarchical scheduling to support network slicing in cellular networks

We study a learning-based hierarchical scheduling framework in support of network slicing for cellular networks. This addresses settings where users and/or service classes are grouped into slices, and resources are allocated hierarchically. The hierarchy is implemented by combining a slice-level scheduler which allocates resources to slices, and flow-level schedulers within slices which opportunistically allocate resources to users/services. Optimizing the slice-level scheduler to maximize system utility is typically hard due to underlying heterogeneity and uncertainty in user channels and performance requirements. We address this by reformulating the problem as an online black-box optimization where slice-level schedulers (parameterized by a weight vector) combined with flow-level schedulers result in user/service level stochastic rewards representing performance fitness; the goal is to learn the best weight vector. We develop a bandit algorithm based on queueing cycles by building on Hierarchical Optimistic Optimization (HOO). The algorithm guides the system to improve the choice of the weight vector based on observed rewards. Theoretical analysis of our algorithm shows a sub-linear regret with respect to an omniscient genie. Finally through simulations, we show that the algorithm adaptively learns the optimal weight vectors when combined with opportunistic and/or utility-maximizing flow-level schedulers.

Derivative-free mixed binary necklace optimization for cyclic-symmetry optimal design problems

This paper presents an adapted trust-region method for computationally expensive black-box optimization problems with mixed binary variables that involve a cyclic symmetry property. Mixed binary problems occur in several practical optimal design problems, e.g., aircraft engine turbines, mooring lines of offshore wind turbines, electric engine stators and rotors. The motivating application for this study is the optimal design of helicopter bladed disk turbomachines. The necklace concept is introduced to deal with the cyclic symmetry property, and to avoid costly black-box objective-function evaluations at equivalent solutions. An adapted distance is proposed for the discrete-space exploration step of the optimization method. A convergence analysis is presented for the trust-region derivative-free algorithm, DFOb-\(d_H\), extended to the mixed-binary case and based on the Hamming distance. The convergence proof is extended to the new algorithm, DFOb-\(d_{neck}\), which is based on the necklace distance. Computational comparison with state-of-the-art black-box optimization methods is performed on a set of analytical problems and on a simplified industrial application.

Data-driven optimization for process systems engineering applications

Most optimization problems in engineering can be formulated as ‘expensive’ black box problems whose solutions are limited by the number of function evaluations. Frequently, engineers develop accurate models of physical systems that are differentiable and/or cheap to evaluate. These models can be solved efficiently, and the solution transferred to the real system. In the absence of gradient information or cheap-to-evaluate models, one must resort to efficient optimization routines that rely only on function evaluations. Creating a model can itself be considered part of the expensive black box optimization process. In this work, we investigate how perceived state-of-the-art derivative-free optimization (DFO) algorithms address different instances of these problems in process engineering. On the algorithms side, we benchmark both model-based and direct-search DFO algorithms. On the problems side, the comparisons are made on one mathematical optimization problem and five chemical engineering applications: model-based design of experiments, flowsheet optimization, real-time optimization, self-optimizing reactions, and controller tuning. Various challenges are considered such as constraint satisfaction, uncertainty, problem dimension and evaluation cost. This work bridges the gap between the derivative-free optimization and process systems literature by providing insight into the efficiency of data-driven optimization algorithms in the process systems domain to advance the digitalization of the chemical and process industries.

Self-tuning portfolio-based Bayesian optimization

Portfolio strategies for Bayesian optimization (BO) aim to mitigate the issue of choosing an acquisition function when performing black-box optimization with Gaussian processes (GP) surrogate models. In that sense, the GP-Hedge is a straightforward portfolio framework commonly used in practice. Our work proposes to overcome existing limitations on the GP-Hedge and related methods, such as reducing the influence of far past evaluations and promoting better exploration. Moreover, we aim to achieve such improvements without sacrificing the practicality of simpler portfolio strategies. More specifically, we propose a new BO method equipped with the aforementioned enhancements enabled by additional self-tuned hyperparameters, which are sampled during the optimization via Thompson sampling. We are able to update the posteriors analytically at each iteration by carefully choosing meaningful conjugate priors. The new approach, named Self-Tuning Portfolio-based BO (SETUP-BO), improves standard portfolio strategies without the need for manually tuning hyperparameters, which preserves easiness of use. We evaluate our method and its competitors in the task of hyperparameter optimization (HPO), a critical step towards automated machine learning (AutoML), following a thorough meta-surrogate benchmarking approach. We also consider a real-world scenario related to the task of fault detection in energy plants. Our methodology achieves promising results, which indicates the viability of the proposed SETUP-BO.

COVID-19: Estimation of the transmission dynamics in Spain using a stochastic simulator and black-box optimization techniques

Background and objectives: Epidemiological models of epidemic spread are an essential tool for optimizing decision-making. The current literature is very extensive and covers a wide variety of deterministic and stochastic models. However, with the increase in computing resources, new, more general, and flexible procedures based on simulation models can assess the effectiveness of measures and quantify the current state of the epidemic. This paper illustrates the potential of this approach to build a new dynamic probabilistic model to estimate the prevalence of SARS-CoV-2 infections in different compartments.Methods: We propose a new probabilistic model in which, for the first time in the epidemic literature, parameter learning is carried out using gradient-free stochastic black-box optimization techniques simulating multiple trajectories of the infection dynamics in a general way, solving an inverse problem that is defined employing the daily information from mortality records.Results: After the application of the new proposal in Spain in the first and successive waves, the result of the model confirms the accuracy to estimate the seroprevalence and allows us to know the real dynamics of the pandemic a posteriori to assess the impact of epidemiological measures by the Spanish government and to plan more efficiently the subsequent decisions with the prior knowledge obtained.Conclusions:The model results allow us to estimate the daily patterns of COVID-19 infections in Spain retrospectively and examine the population’s exposure to the virus dynamically in contrast to seroprevalence surveys. Furthermore, given the flexibility of our simulation framework, we can model situations —even using non-parametric distributions between the different compartments in the model— that other models in the existing literature cannot. Our general optimization strategy remains valid in these cases, and we can easily create other non-standard simulation epidemic models that incorporate more complex and dynamic structures.

A competitive variable-fidelity surrogate-assisted CMA-ES algorithm using data mining techniques

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm is one of the most successful continuous black-box optimization algorithms widely used in many fields. However, the algorithm would be computationally unaffordable due to the sharp increase in dimension and complexity of problems. To improve the computational efficiency and global optimizing ability, a competitive variable-fidelity surrogate-assisted CMA-ES (CVFS-CMA-ES) algorithm using data mining techniques is proposed in this paper. In the first data mining of CVFS-CMA-ES, the establishment method of the competitive variable-fidelity model (VFM) based on the fuzzy clustering algorithm is developed, aiming at making full use of high-fidelity information and focusing more on potential areas of design space. In the second data mining of CVFS-CMA-ES, the Lower Confidence Bound (LCB) method is modified by introducing the step size of CMA-ES to control the uncertain term adaptively, and the sample points are screened in each generation of CMA-ES based on the modified LCB method, which can make a balance between exploration and exploitation in the optimization process. The performance of CVFS-CMA-ES is compared with four known black-box optimization algorithms. Firstly, ten numerical examples of 10-dimensional and 30-dimensional benchmark functions are carried out, respectively. Moreover, a 20-dimensional engineering example of the aerospace variable-stiffness composite shell under combined loadings is studied in detail. Results of benchmark functions and engineering examples verify the high efficiency, robustness and high global optimizing ability of the proposed CVFS-CMA-ES in comparison to other algorithms.

Optimal Controller Design for Transient Stability Enhancement of Grid-Following Converters Under Weak-Grid Conditions

Modeling and design-oriented control of transient stability of grid-following converters have attained an increasing interest in recent years. Despite novel nonlinear models enabling a design-oriented enhanced transient stability controller, the focus has so far been limited to study only the synchronization dynamics of the phase-locked loop. To expand upon the knowledge of the large-signal performance and stability, this article proposes a systematic analysis procedure of a grid-following converter under weak-grid conditions and large-signal disturbances including the outer dc-link and ac-side voltage control loops. A reduced-order large-signal model is used to analyze the large-signal nonlinear behavior of the system using the area of the basin of attraction as a measure for large-signal robustness. Here, stabilizing and destabilizing trends for outer-loop controller parameters are given. Through a surrogate-model expensive black-box optimization algorithm, a computational-efficient optimal design of the outer-loop controller parameters is proposed to maximize the large-signal robustness. Finally, a recommendation and a design guideline for converter constraints and outer-loop controller parameters are given. This can be used to identify the influencing parameters for grid-following converters under large-signal disturbances, and as a tool for fast controller optimization toward large-signal robustness.

Black-Box Function Aerodynamic Topology Optimization Algorithm via Machine Learning Technologies

The objective of this research is to construct an efficient global topology optimization method using machine learning technologies. In the conventional design process of mechanical design, the conceptual design is the earliest stage of the design process; and it is carried out based on the designer’s own idea/experience. Therefore, it is difficult to obtain innovative and high-quality concepts overwhelming the designer’s knowledge since the earliest stage of the design process has the largest impact. In this research, therefore, the black-box function aerodynamic topology optimization algorithm via machine learning technologies (FANTOM) is developed to overcome the problem. In the FANTOM approach, topology optimization problems are solved using/combining two efficient global optimization methods developed by the authors: the efficient global optimization method for discontinuous optimization problems with infeasible regions using classification method, and the efficient global optimization method via clustering/classification methods and exploration strategy. In the present approach, topological optimal designs can be obtained only by setting an objective function and constraint conditions. The validity of the FANTOM approach is demonstrated in an inviscid drag minimization problem at a two-dimensional supersonic flow condition, which provides an optimal topology as the Busemann biplane airfoil. Executing topology optimizations with the variation in freestream Mach number, it is also demonstrated that the FANTOM approach can explore topological optimal designs robustly.

Advances in meta-heuristic methods for large-scale black-box optimization problems

The optimal design of complex engineering equipment usually faces high-complexity, high-dimensional optimization problems – the so-called black-box optimization problems (LBOPs) – which are characterized by unavailable mathematical expressions of objective functions and/or constraint functions, and high dimensionality of design variables. The LBOPs have attracted the interest of scholars in various fields in recent years, and meta-heuristic algorithms are considered effective methods for solving these problems. This paper comprehensively summarizes recent research progress in meta-heuristic algorithms for solving LBOPs, including meta-heuristic algorithms with and without decomposition strategies, and meta-heuristic algorithms for handling computationally expensive large-scale optimization problems. Finally, possible future research directions of meta-heuristic methods for solving LBOPs are proposed.

A note on the multiplicative fairness score in the NIJ recidivism forecasting challenge

BackgroundThe 2021 NIJ recidivism forecasting challenge asks participants to construct predictive models of recidivism while balancing false positive rates across groups of Black and white individuals through a multiplicative fairness score. We investigate the performance of several models for forecasting 1-year recidivism and optimizing the NIJ multiplicative fairness metric.MethodsWe consider standard linear and logistic regression, a penalized regression that optimizes a convex surrogate loss (that we show has an analytical solution), two post-processing techniques, linear regression with re-balanced data, a black-box general purpose optimizer applied directly to the NIJ metric and a gradient boosting machine learning approach.ResultsFor the set of models investigated, we find that a simple heuristic of truncating scores at the decision threshold (thus predicting no recidivism across the data) yields as good or better NIJ fairness scores on held out data compared to other, more sophisticated approaches. We also find that when the cutoff is further away from the base rate of recidivism, as is the case in the competition where the base rate is 0.29 and the cutoff is 0.5, then simply optimizing the mean square error gives nearly optimal NIJ fairness metric solutions.ConclusionsThe multiplicative metric in the 2021 NIJ recidivism forecasting competition encourages solutions that simply optimize MSE and/or use truncation, therefore yielding trivial solutions that forecast no one will recidivate.

Impact of the deployment of solar photovoltaic and electrical vehicle on the low voltage unbalanced networks and the role of battery energy storage systems

The deployment of low carbon technologies (LCTs) such as solar photovoltaics (PVs) and electric vehicles (EVs) is increasing due to their various benefits. However, their rapid integration in the residential networks imposes critical technical issues to the network operators. In this paper, a sensitivity analysis is conducted to investigate the impact of these technologies on low voltage unbalanced residential networks located in Northern Ireland. Different penetrations are investigated, and the results are assessed using various technical indices. The simulations are performed using actual load measurements in a 10-minute resolution obtained from the network operator of Northern Ireland. The impact of LCTs on the phase imbalance is considered by analyzing the voltage unbalance and neutral losses. The results suggest that monitoring each feeder and phase separately and consider the voltage unbalance in the assessment as it can be violated without a voltage magnitude violation occurring, in addition, the farthest nodes from the transformer bus were found to be very sensitive to violations. The solution-based community battery energy storage systems (BESS) is investigated by introducing sizing and scheduling algorithms to solve the network issues and enhance the network performance using black-box optimization. The results proved the wide capability of the community BESS in providing the network with different services. The main purpose of this study is to provide a full understanding of the future challenges posed by the deployment of LCTs and the role of BESS in accelerating the transition towards a low carbon future by supporting the network operation.

Risk-based functional black-box optimization: Contribution to the NASA Langley UQ challenge on optimization under uncertainty

This paper presents an approach to solve the 2019/2020 NASA Langley UQ challenge problem on optimization under uncertainty. We define an uncertainty model (UM) as a pair fa|e,E, where fa|e is a probability density over a for each e∈E, and proceed to infer fa|e in a Bayesian fashion. Special attention is given to dimensionality reduction of the functional (time-series) data, to obtain a finite dimensional representation suitable for robust Bayesian inversion. Reliability analysis is performed using fa|e, whereas for design optimization we approximate fa|e using truncated Gaussians and a Gaussian copula. We apply an unscented transform (UT) in the standard normal space to estimate moments of the limit state, which is numerically very efficient. Design optimization is performed with this procedure to obtain negligible failure probability in g1 and g3 and acceptable failure probability and severity in g2.

CS-DE: Cooperative Strategy based Differential Evolution with population diversity enhancement

Differential Evolution (DE) is one of the most effective approaches in Evolutionary Computation to tackling complex black-box optimization, however, the overall performance of a DE variant is heavily dependent on its mutation strategy and the associated parameter control schemes. According to the “No free Lunch theorem”, every mutation strategy has its own fatal defect, therefore, DE variant with a single mutation strategy can not tackle all optimization problems. Furthermore, DE variant with ensemble of mutation strategies usually needs more parameter control schemes, which inevitably decreases the practicality of it. Here in this paper, a novel Cooperative Strategy based DE (CS-DE) with population diversity enhancement is proposed, and the CS-DE algorithm has the following contributions: First, a cooperative strategy pool containing two similar mutation strategies is advanced in the algorithm, and the two mutation strategies share the same parameter control. Second, a novel grouping strategy is proposed, then the two mutation strategies can be chosen in an adaptive manner by different individuals in generating trial vectors during the evolution. Third, novel adaptation schemes for the three control parameters F,CR and PS as well as a stagnation based re-initialization mechanism are also proposed in our CS-DE algorithm. 58 benchmarks from the CEC2013 and CEC2014 test suites are employed in algorithm validation, and experiment results show the competitiveness of the CS-DE algorithm with several state-of-the-art DE variants.

Multi-fidelity black-box optimization for time-optimal quadrotor maneuvers

We consider the problem of generating a time-optimal quadrotor trajectory for highly maneuverable vehicles, such as quadrotor aircraft. The problem is challenging because the optimal trajectory is located on the boundary of the set of dynamically feasible trajectories. This boundary is hard to model as it involves limitations of the entire system, including complex aerodynamic and electromechanical phenomena, in agile high-speed flight. In this work, we propose a multi-fidelity Bayesian optimization framework that models the feasibility constraints based on analytical approximation, numerical simulation, and real-world flight experiments. By combining evaluations at different fidelities, trajectory time is optimized while the number of costly flight experiments is kept to a minimum. The algorithm is thoroughly evaluated for the trajectory generation problem in two different scenarios: (1) connecting predetermined waypoints; (2) planning in obstacle-rich environments. For each scenario, we conduct both simulation and real-world flight experiments at speeds up to 11 m/s. Resulting trajectories were found to be significantly faster than those obtained through minimum-snap trajectory planning.

Natural evolutionary strategies for variational quantum computation

Natural evolutionary strategies (NES) are a family of gradient-free black-box optimization algorithms. This study illustrates their use for the optimization of randomly initialized parameterized quantum circuits (PQCs) in the region of vanishing gradients. We show that using the NES gradient estimator the exponential decrease in variance can be alleviated. We implement two specific approaches, the exponential and separable NES, for parameter optimization of PQCs and compare them against standard gradient descent. We apply them to two different problems of ground state energy estimation using variational quantum eigensolver and state preparation with circuits of varying depth and length. We also introduce batch optimization for circuits with larger depth to extend the use of ES to a larger number of parameters. We achieve accuracy comparable to state-of-the-art optimization techniques in all the above cases with a lower number of circuit evaluations. Our empirical results indicate that one can use NES as a hybrid tool in tandem with other gradient-based methods for optimization of deep quantum circuits in regions with vanishing gradients.